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312                       ANALYTICAL MIOCI IAN ICS
the moment of inertia of the pendulum equals ml?. Therefore substituting this value of / in the expressions for /\> and P and replacing D by I we obtain
™ my I)
(XV) for the first approximation, and
for the second approximation.
243. Equivalent Simple Pendulum.. — A simple pendulum which has the same period as a physical pendulum is (tailed the equivalent simple pendulum of the latter. If / denotes the length of the equivalent simple pendulum, then
a                  y>
ft"2  J~  7)2
.,    Z_A_±«-.                                  (XVI)
For a given value of D and a given direction of the axis, K is constant. Therefore if the direction of the axis is not •changed I is a function of I) alone. If we plot the last equation with I as ordinate and I) as abscissa we obtain a curve •similar to that of Fig. 139. It is evident from the curve that the value of I is infinitely large for I) = 0, but it diminishes
rapidly to the minimum value—    as I) reaches the value A". As D is increased further I increases continually.  It will be ob-
r>  rr
served that for a given value of I greater than      there are two values of I), one of which is less and the other greater than K.